# Properties

 Label 2112.2.a.a Level $2112$ Weight $2$ Character orbit 2112.a Self dual yes Analytic conductor $16.864$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2112.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.8644049069$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 264) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} - 4q^{5} - 2q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - 4q^{5} - 2q^{7} + q^{9} + q^{11} + 4q^{15} - 6q^{17} - 4q^{19} + 2q^{21} - 6q^{23} + 11q^{25} - q^{27} - 6q^{29} - q^{33} + 8q^{35} - 6q^{37} - 10q^{41} + 8q^{43} - 4q^{45} + 6q^{47} - 3q^{49} + 6q^{51} + 12q^{53} - 4q^{55} + 4q^{57} + 8q^{59} - 4q^{61} - 2q^{63} + 12q^{67} + 6q^{69} + 10q^{71} + 2q^{73} - 11q^{75} - 2q^{77} + 2q^{79} + q^{81} - 12q^{83} + 24q^{85} + 6q^{87} - 6q^{89} + 16q^{95} + 14q^{97} + q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 −4.00000 0 −2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.a.a 1
3.b odd 2 1 6336.2.a.ck 1
4.b odd 2 1 2112.2.a.p 1
8.b even 2 1 264.2.a.d 1
8.d odd 2 1 528.2.a.f 1
12.b even 2 1 6336.2.a.cn 1
24.f even 2 1 1584.2.a.a 1
24.h odd 2 1 792.2.a.a 1
40.f even 2 1 6600.2.a.k 1
40.i odd 4 2 6600.2.d.c 2
88.b odd 2 1 2904.2.a.o 1
88.g even 2 1 5808.2.a.p 1
264.m even 2 1 8712.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.d 1 8.b even 2 1
528.2.a.f 1 8.d odd 2 1
792.2.a.a 1 24.h odd 2 1
1584.2.a.a 1 24.f even 2 1
2112.2.a.a 1 1.a even 1 1 trivial
2112.2.a.p 1 4.b odd 2 1
2904.2.a.o 1 88.b odd 2 1
5808.2.a.p 1 88.g even 2 1
6336.2.a.ck 1 3.b odd 2 1
6336.2.a.cn 1 12.b even 2 1
6600.2.a.k 1 40.f even 2 1
6600.2.d.c 2 40.i odd 4 2
8712.2.a.a 1 264.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2112))$$:

 $$T_{5} + 4$$ $$T_{7} + 2$$ $$T_{13}$$ $$T_{17} + 6$$ $$T_{19} + 4$$ $$T_{23} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T$$
$5$ $$1 + 4 T + 5 T^{2}$$
$7$ $$1 + 2 T + 7 T^{2}$$
$11$ $$1 - T$$
$13$ $$1 + 13 T^{2}$$
$17$ $$1 + 6 T + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 6 T + 23 T^{2}$$
$29$ $$1 + 6 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 + 6 T + 37 T^{2}$$
$41$ $$1 + 10 T + 41 T^{2}$$
$43$ $$1 - 8 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 - 12 T + 53 T^{2}$$
$59$ $$1 - 8 T + 59 T^{2}$$
$61$ $$1 + 4 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 - 10 T + 71 T^{2}$$
$73$ $$1 - 2 T + 73 T^{2}$$
$79$ $$1 - 2 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 6 T + 89 T^{2}$$
$97$ $$1 - 14 T + 97 T^{2}$$